I'd run it through a mild washload with a load of sweaters or fluffy blankets -- something that's not heavy when wet and could get wrapped around it (like towels or jeans). Purex detergent with Febreeze deodorizer is wonderful stuff for removing odors!
When it goes to spin dry after the rinse cycle, stop the machine a few times and reposition the bear so that it won't be 'warped' to one side or something. Tumble it in a hot dryer with the load of laundry (in a pillow case as someone suggested, maybe) just long enough to get it good and warm and take it out. Either repeat this 'get it hot, take it out' procedure til it dries (instead of tumbling it to death) or lay it in the sun, if possible.
Even if you can just rinse the side that got 'wet' with water and white vinegar (hold it higher than the spray of water so that you only spray the part that got wet) and let the water drain back out of it a couple times (instead of getting it wet through and through), then dry it in the sun, that would help immensely. The sun and wind help deodorize, bleach, AND dry things. Also, if you have a 'Sham-Wow' or some microfiber cloths (like terrycloth), squeeze it with them (much better than terry towels for absorbing liquid!)